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f8d4060b3300edc5bf1ab29d53f4c6e4d531d703 | subsection | 1,098 | 1,121 | Equivariant spaces | Given a based H-space Z, the H-equivariant collapse mapl_Z \ : \ G\ltimes _H Z\ \longrightarrow \ Z\wedge S^Lwas introduced in (REF ); here L=T_{e H}(G/H)
is the tangent H-representation.
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610065654d89d4e57afa299aa840047c2ed2376c | subsection | 1,099 | 1,121 | Enriched functor categories | In this final appendix we review definitions, properties and constructions
involving categories of enriched functors.
The general setup consists of:a complete and cocomplete closed symmetric monoidal category {\mathcal {V}}
(the `base category').
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1e5dd02999e98e093ddbc9a42a1014f66ad6a0da | subsection | 1,100 | 1,121 | Enriched functor categories | The (d,e)-component of the morphism \alpha is the composite{\prod }_c \ \underline{{\mathcal {V}}}(X(c),Y(c)) \ \xrightarrow{} \ \underline{{\mathcal {V}}}(X(d),Y(d)) \ \xrightarrow{} \ \underline{{\mathcal {V}}}({\mathcal {D}}(d,e)\otimes X(d),Y(e))where the second morphism is adjoint to\underline{{\mathcal {V}}}(X(d)... | {
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2178ab5ceb5079e16603f44ba29aab918521f826 | subsection | 1,101 | 1,121 | Enriched functor categories | When applied to the defining equalizer
for \underline{{\mathcal {D}}}^\ast (X,Y) this shows that
the set of morphisms {\mathcal {D}}^\ast (X,Y) in the functor category {\mathcal {D}}^\ast
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a6b309e5ea391c99acb39f963a951767627da416 | subsection | 1,102 | 1,121 | Enriched functor categories | For a {\mathcal {V}}-functor Y in {\mathcal {D}}^{\ast },
the evaluation morphism is the composite\underline{{\mathcal {D}}}^{\ast }(d^\ast ,Y)\ \xrightarrow{} \ \underline{{\mathcal {V}}}(d^\ast (d),Y(d)) \ &= \ \underline{{\mathcal {V}}}({\mathcal {D}}(d,d),Y(d)) \\
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14a3a898367bfe095965de10545ee4a02088f070 | subsection | 1,103 | 1,121 | Enriched functor categories | This yields the box product of global functors.We denote by {\mathcal {D}}\otimes {\mathcal {D}} the {\mathcal {V}}-category whose objects are
pairs of {\mathcal {D}}-objects, and with morphism {\mathcal {V}}-objects({\mathcal {D}}\otimes {\mathcal {D}})((d,d^{\prime }),(e,e^{\prime }))\ = \ {\mathcal {D}}(d,e)\otimes ... | {
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c5b506bb718bfe4c7712409e015675acc7592cd8 | subsection | 1,104 | 1,121 | Enriched functor categories | Such a Kan extension exists because {\mathcal {V}} is cocomplete
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635e48226724daae0ad9842766120d0767895fb9 | subsection | 1,105 | 1,121 | Enriched functor categories | Indeed, if (X\Box Y,i) and (X\Box ^{\prime } Y,i^{\prime })
are two box products, then the universal properties provide unique morphisms
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98ca017c511e7178d394736872bc1e5f98345086 | subsection | 1,106 | 1,121 | Enriched functor categories | So we make the following conventions:(Right unit) We choose X\Box 0^\ast =X with universal bimorphism
i:(X,0^\ast )\longrightarrow X given by the maps
X(d)\otimes 0^\ast (e)\ = \ X(d)\otimes {\mathcal {D}}(0,e)\ \xrightarrow{} \ &X(d)\otimes {\mathcal {D}}(d\oplus 0,d\oplus e)\\
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060dc721428a0f62be5d5203c2866326bb38ab80 | subsection | 1,107 | 1,121 | Enriched functor categories | The uniqueness of representing objects gives a unique isomorphism
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d7facd924d7f500695a49db952262d3f336f1701 | subsection | 1,108 | 1,121 | Enriched functor categories | The two types of associativity isomorphisms
X\Box (Y\otimes A)\cong (X\Box Y)\otimes A and
Y\otimes (A\otimes B)\cong (Y\otimes A)\otimes B
are compatible in the sense of a commuting pentagon:@C=-5mm{
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243fd5a39dfa571250f7b6ef8d637e518b8c1944 | subsection | 1,109 | 1,121 | Enriched functor categories | The universal properties of the convolution product and
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are inverse to each other.Remark 4.159 (Internal function objects)
The box product is a closed monoidal product
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8ea5dd2d46ebed7c526809922ad9eb107b13c24b | subsection | 1,110 | 1,121 | Enriched functor categories | The extension l_m(Z) of an enriched functor Z:{\mathcal {D}}_{\le m}\longrightarrow {\mathcal {V}}
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8511cafcec97aabc99cde89c37bab2216d501f7a | subsection | 1,111 | 1,121 | Enriched functor categories | The m-th latching objectlatching object!of an enriched functorL_m Y - m-th latching object of Y
of Y is the {\mathcal {D}}(m)-objectL_m Y \ = \ (\operatorname{sk}^{m-1} Y)(\mathbf {m}) \ ;it comes with a natural {\mathcal {D}}(m)-equivariant morphism\nu _m=i_{m-1}(\mathbf {m})\ :\ L_m Y\ \longrightarrow \ Y(\mathbf {m}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4a7705fd518e5f39fc055f4e158688b372dab328 | subsection | 1,112 | 1,121 | Enriched functor categories | For n>m the latching morphism
\nu _n:L_n({\mathcal {D}}(d,-)\otimes A) \longrightarrow {\mathcal {D}}(d,\mathbf {n})\otimes A
is an isomorphism. So the skeleton \operatorname{sk}^n ({\mathcal {D}}(d,-)\otimes A) is initial for n<m
and \operatorname{sk}^n ({\mathcal {D}}(d,-)\otimes A)={\mathcal {D}}(d,-)\otimes A is th... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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059a0af0fa416d13e315f4848beed10ed7078ab7 | subsection | 1,113 | 1,121 | Enriched functor categories | Then the square (REF ) evaluates to@C=25mm{
{\mathcal {D}}(\mathbf {m},\mathbf {m})\otimes _{{\mathcal {D}}(m)} L_m Y
[r]^{{\mathcal {D}}(\mathbf {m},\mathbf {m})\otimes _{{\mathcal {D}}(m)}\nu _m} [d] &
{\mathcal {D}}(\mathbf {m},\mathbf {m})\otimes _{{\mathcal {D}}(m)} Y(\mathbf {m}) [d]\\
L_m Y [r]_-{j_m(\mathbf {m}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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dcae323574f25556d7a21f88c7bc465847fae683 | subsection | 1,114 | 1,121 | Enriched functor categories | The original morphism f:A\longrightarrow B
factors as the composite of the countable sequenceA=\operatorname{sk}^{-1}[f]\ \xrightarrow{}\ \operatorname{sk}^0[f] \ \xrightarrow{}\ \operatorname{sk}^1[f] \ \longrightarrow \ \cdots \ \xrightarrow{}\ \operatorname{sk}^m [f] \ \longrightarrow \ \cdots \ .If d has dimension ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a07c26f49c0c5e2551dfff6eac98dfbea167bf53 | subsection | 1,115 | 1,121 | Enriched functor categories | If the pair(\nu _m i=i(\mathbf {m})\cup \nu _m^B:A(\mathbf {m})\cup _{L_m A}L_m B\longrightarrow B(\mathbf {m}),\quad f(\mathbf {m}):X(\mathbf {m})\longrightarrow Y(\mathbf {m}))has the lifting property in the category of {\mathcal {D}}(m)-objects for
every m\ge 0, then the pair (i,f) has the lifting property
in the fu... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6eec6ef200235fff9bde9210fc4caed6e5945b20 | subsection | 1,116 | 1,121 | Enriched functor categories | We call a morphism f:X\longrightarrow Y in {\mathcal {D}}^\asta level equivalence if f(\mathbf {m}):X(\mathbf {m})\longrightarrow Y(\mathbf {m})
is a weak equivalence in the model structure {\mathcal {C}}(m) for all m\ge 0;
a level fibration if the morphism
f(\mathbf {m}):X(\mathbf {m})\longrightarrow Y(\mathbf {m}) i... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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fc2bbd1b884d3853a602cd5a991ef0a15d4c0207 | subsection | 1,117 | 1,121 | Enriched functor categories | Suppose that the fibrations in the model structure {\mathcal {C}}(m)
are detected by a set of morphisms J(m);
then the level fibrations are detected by the set of morphisms
\lbrace G_m j \ | \ m\ge 0,\, j\in J(m)\rbrace \ .
Similarly, if the acyclic fibrations in the model structure {\mathcal {C}}(m)
are detected by a... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5faebce37f4e7b99fc2b160aa7437dd467139864 | subsection | 1,118 | 1,121 | Enriched functor categories | The pushout square (REF )
in level m+n is a pushout of {\mathcal {D}}(m+n)-objects@C=20mm{
{\mathcal {D}}(\mathbf {m},\mathbf {m+n})\otimes _{{\mathcal {D}}(m)}(A(\mathbf {m})\cup _{L_m A}L_m B)
[r]^-{{\mathcal {D}}(\mathbf {m},\mathbf {m+n})\otimes _{{\mathcal {D}}(m)} (\nu _m i)} [d]
& {\mathcal {D}}(\mathbf {m},\mat... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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761f58207efe5f3bb3d1255f74803f5a50178b8a | subsection | 1,119 | 1,121 | Enriched functor categories | Then we have all the data necessary to define the
m-th latching object L_m B; moreover, the `partial morphism'
q:B\longrightarrow X provides a {\mathcal {D}}(m)-morphism L_m B\longrightarrow X(\mathbf {m})
such that the square{L_m A [r]^-{L_m i} [d]_{\nu _m^A} & L_m B [d] \\
A(\mathbf {m}) [r]_-{f(\mathbf {m})} & X(\ma... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e177a259b73de10f73b8c832210e17a6aebe9b29 | subsection | 1,120 | 1,121 | Enriched functor categories | In each of the model structures {\mathcal {C}}(m)
the cofibrations have the left lifting property with respect
to the acyclic fibrations; so by
Proposition REF
the cofibrations in {\mathcal {D}}^\ast have the left lifting property
with respect to level equivalences which are also level fibrations.We postpone the proof... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e752dc892ff8475befee1fbb25ccf2594ba6ea37 | abstract | 0 | 31 | Abstract | We consider the problem of counting motifs in bipartite affiliation networks,
such as author-paper, user-product, and actor-movie relations. We focus on
counting the number of occurrences of a "butterfly", a complete $2 \times 2$
biclique, the simplest cohesive higher-order structure in a bipartite graph.
Our main cont... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
"Ahmet Erdem Sariyuce",
"Srikanta Tirthapura"
] | [
"cs.DM"
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fdbdd594835d05183d163329fc1459d0e6832149 | subsection | 1 | 31 | Introduction | Graph motifs are used to model and examine interactions among small sets of vertices in networks. Finding frequent patterns of interactions can reveal functions of participating entities , , , , , and help characterize the network. Also known as graphlets or higher-order structures, motifs are regarded as basic buildin... | {
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"raw": "authorpersonC. Seshadhri, personA. Pinar, and personT. G. Kolda. year2014. Triadic Measures on Graphs: The Power of Wedge Sampling. journalStatistical Analysis and Da... | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
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cad662b85b5f1fff473fe25b4edc675481ce6663 | subsection | 2 | 31 | Introduction | Instead, natural motifs in a bipartite network are bicliques of small size.The most basic motif that models cohesion in a bipartite network is the complete 2 \times 2 biclique, also known as a butterfly , or a rectangle . Although there have been attempts at defining other cohesive motifs in bipartite networks, such as... | {
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"openalex_id": "https://openalex.org/W2770225549",
"raw": "authorpersonAhmet Erdem Sarıyüce and personAli Pinar. year2018. Peeling Bipartite Networks for Dense Subgraph Discovery. In booktitleWSDM.",
... | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
"Ahmet Erdem Sariyuce",
"Srikanta Tirthapura"
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a8dcd4737be320d9ce22073a7eb2a148613c60d4 | subsection | 3 | 31 | Introduction | These results show that the algorithms can handle massive graphs with hundreds of millions of edges and trillions of butterflies.
Our most efficient sampling algorithm, which we call ESamp+Fast-eBFC, gives estimates with a relative error less than 1 percent within 5 seconds, even for large graphs with trillions of butt... | {
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23dd957e248668c8f423b1d5d165e8185f2c4e1a | subsection | 4 | 31 | Preliminaries | We consider simple, unweighted, bipartite graphs, where there are no self-loops or multiple edges between vertices.
Let G = (V, E) be a simple bipartite graph with n=|V| vertices and m=|E| edges.
Vertex set V is partitioned into two sets L and R such that V = L \cup R and L \cap R = \emptyset .
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fbbaa9aad4cdf37e2ab157ce5f6a45a108a1b6f9 | subsection | 5 | 31 | Preliminaries | For parameters \epsilon , \delta \in [0, 1], an (\epsilon , \delta )-{}{}approximation of a number Z is a random variable \hat{Z} such that \Pr [|\hat{Z} - Z| > \epsilon Z] \le \delta .
}\begin{}[t!]
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e38f014c95010bf4806035634e7bd0b1ab32c5e5 | subsection | 6 | 31 | Preliminaries | \end{}\begin{}[t!]
\small \centering {}{!}{
\begin{}{|c| @{{}} |r|r|r|r|r|r|}
\hline Bipartite graph & |L| & |R| & |E| & \sum _{\ell \in {}L}{d_\ell ^2} & \sum _{r\in {}R}{d_r^2} & \mathbin { \\ \hline \left( 10^4,10 \right)-biclique & {10000} & {10} & {100000} & {1000000} & {1000000000} & {2249775000} \\ \hline \href ... | {
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790f09b1b4ee508684dd11659009992b664ed0c2 | subsection | 7 | 31 | Preliminaries | L and R are vertex partitions, E is the edge set. The sum of degree squares for L and R, and the number of butterflies are shown.
}
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\section {Related Work}
}}\textbf {Bipartite graph motifs:} Modeling the smallest unit of cohesion enables a principled way to analyze network... | {
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bcbe0a065e18480cdb0d1a3bab70c5253e25fcdf | subsection | 8 | 31 | Exact Butterfly Counting | We first present the basic equation for the number of butterflies in a bipartite graph G and the base (state-of-the-art) algorithm by Wang et al. that implements the equation.Lemma 1For a bipartite graph G = (V = (L \cup R), E),We have two observations about Equation (3) and algo:exactBFC.
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ed4485e7bbd283a865293524054bc4b61ecb0a56 | subsection | 9 | 31 | Performance of Exact Butterfly Counting | We compare the runtime of our algorithm (ExactBFC) with Wang et al. (WFC).
From our theoretical analysis in sec:exact, our algorithm is expected to be faster than WFC.
figure:exactVSwang shows a comparison of the runtimes of the two algorithms. We note the following points.
(1) ExactBFC is always faster than WFC. This... | {
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ded75a361b86fdc24483b7b688b5ca7679082921 | subsection | 10 | 31 | Local Butterfly Counting | We present two algorithms for local butterfly counting, vBFC (algo:local-verBFC) for counting the number of butterflies \mathbin {_v that contain a given vertex v, and {\textsc {eBFC}}~({algo:local-edgBFC}) for counting the number of butterflies \mathbin {_e that contain an edge e. Both algorithms employ procedures sim... | {
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18ce2415f63110d2d63cb92c74313524a13cffa0 | subsection | 11 | 31 | Approximation by Local Sampling | In this section, we present approaches to approximating \mathbin {(G) using random sampling. The intuition behind sampling is to examine a randomly sampled subgraph of G and compute the number of butterflies in the subgraph to derive an estimate of \mathbin {(G). Since the subgraph is typically much smaller than G, it ... | {
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60bd9627cdb5c6f9aad997b6e832bca79b2365be | subsection | 12 | 31 | Vertex Sampling (Algorithm | The idea in VSamp is to sample a random vertex v and count the number of butterflies that contain v – this is accomplished by counting the number of butterflies in the induced subgraph consisting of the distance-2 neighborhood of v in the graph. We show that the algorithm, described in algo:versamp, yields an unbiased ... | {
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afb02ad8d986f7053df39910e34c1104e47fec0f | subsection | 13 | 31 | Vertex Sampling (Algorithm | Triadic closure in two-mode networks: Redefining
the global and local clustering coefficients}.
{\section {Conclusion}
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0.... | |
dd1a556d26721b61dbdb1a33d06856c6443fba77 | subsection | 16 | 31 | Vertex Sampling (Algorithm | Seshadhri},
{and} {person}{\frac{}{ESCAPE: Efficiently Counting All 5-Vertex
Subgraphs}Vaidyanathan Vishal}.}
{year}{bad group22bad group222017}{}.
{
{
Let f : V \rightarrow \lbrace 1,\ldots ,\rbrace *{map to random colors}E^{\prime \bibitem {\left(\@root {\sqrt{\frac{Counting and Sampling Triangles from a Graph
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"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
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"Srikanta Tirthapura"
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7aa913d17e66d04ba7395e97193c36e81460281f | subsection | 17 | 31 | Vertex Sampling (Algorithm | Pavan}, {person}{K. Tangwongsan},
{person}{S. Tirthapura}, {\frac{}{}However, \\textsc \{ESpar\}\\mml@font@normal {}~has other downsides when compared with \\textsc \{ESamp\}\\mml@font@normal {}.
First, the memory consumption of \\textsc \{ESpar\}\\mml@font@normal {}~is O(mp) where p is a parameter, and is larger than ... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
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5434ca10dffd6f933196c37718776cf97e4abee0 | subsection | 18 | 31 | Vertex Sampling (Algorithm | Overall, two algorithms take similar times to reach a 1\% error on all graphs we considered, in the range of 1.7-5 sec, with \\textsc \{ESpar\}\\mml@font@normal {}~achieving this accuracy faster than \\textsc \{ESamp\}\\mml@font@normal {}~with \\textsc \{Fast-eBFC\}\\mml@font@normal {}.
journal}{\\emph \{{\left(\@root ... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
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2e2a45bc6c98ba81ded915f8ed7ffc6a255d9be2 | subsection | 19 | 31 | Vertex Sampling (Algorithm | To understand this, we begin with {lem:edge-spars,lem:clr-spars} which show expressions bounding the variances in terms of p_{1w}, p_{1e}, and p_{2v}, also summarized in {table:variance-sprs}. The difference in the variance between \textsc {ClrSpar}{}~ and \textsc {ESpar}{}~ boils down to (\mathbin {{}p^{-3}+p_{2v\bibi... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
"Ahmet Erdem Sariyuce",
"Srikanta Tirthapura"
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d03c0f08ff2c7bc84cf4722f0b3f7be2981d6115 | subsection | 20 | 31 | Vertex Sampling (Algorithm | However, as shown in {figure:jrn-sprs-err-prob,figure:web-sprs-err-prob,figure:wiki-sprs-err-prob}, \textsc {ClrSpar}{}~{}requires a larger sampling probability to achieve a reasonable accuracy.
\bibitem {bad group22bad group22Lim15}
{author}{{\begin{}{31}
\end{}person}{Yongsub Lim} {and} {\bibliographystyle {plain}
pe... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
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91d68aa3bef520097ceecd07a4f273e1d83992d4 | subsection | 21 | 31 | Vertex Sampling (Algorithm | Wu, personK. Yi, and
personZ. Li. year2016.
Counting Triangles in Large Graphs by Random
Sampling.
journalIEEE TKDE volume28,
number8 (year2016), pages2013–2026.The idea in VSamp is to sample a random vertex v and count the number of butterflies that contain v – this is accomplished by counting the number of butterflie... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
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c26f8c97097c1f8cc4427f6882de799a830075c0 | subsection | 22 | 31 | Vertex Sampling (Algorithm | Our best sampling and sparsification algorithms yield less than 1\% relative error within \approx 5 and \approx 4 seconds for all the networks we considered, whereas the state-of-the-art exact algorithm does not complete even in 40,000 secs on the \\texttt \\mml@font@monospace \{Web\}\\mml@font@normal ~graph.
Triadic c... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
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b6951124986db44251810a580cf7c84e82ab4c53 | subsection | 23 | 31 | Vertex Sampling (Algorithm | Alon}.}
{year}{2002}{}.
{Network Motifs: Simple Building Blocks of Complex
Networks}.
{journal}{\\emph \{Science\}\} {volume}{298},
{number}{5594} ({year}{2002}), {pages}{824--827}.
} + p_{1e}p^{-1} + p_{2v}p^{-1}} \\\hline \\texttt \\mml@font@monospace \{Deli\}\\mml@font@normal & 4.047 \times 10^{15} & 9.163 \times 10... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
"Ahmet Erdem Sariyuce",
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5c2f2d302be6210421cf3ab349f064dbfb1c3557 | subsection | 24 | 31 | Vertex Sampling (Algorithm | ({year}{2017}).
{({\\tt \\mml@font@monospace https://developers.facebook.com/docs/graph-api}).}
\\emph \{SIAM J. Comput.\}\} {volume}{7},
{number}{4} ({year}{1978}), {pages}{413--423}.
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{author}{{person}{L. De~Stefani}, {person}{A. Epasto},
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} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
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c634a134cbc058c1639578653f1b5a1ed4a874ae | subsection | 25 | 31 | Vertex Sampling (Algorithm | Seshadhri},
{and} {person}{\frac{}{ESCAPE: Efficiently Counting All 5-Vertex
Subgraphs}Vaidyanathan Vishal}.}
{year}{bad group22bad group222017}{}.
{
{
Let f : V \rightarrow \lbrace 1,\ldots ,\rbrace *{map to random colors}E^{\prime \bibitem {\left(\@root {\sqrt{\frac{Counting and Sampling Triangles from a Graph
Stream... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
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67930802f744c0d3e2fc930d24695f20d52f3332 | subsection | 26 | 31 | Vertex Sampling (Algorithm | Pavan}, {person}{K. Tangwongsan},
{person}{S. Tirthapura}, {\frac{}{}However, \\textsc \{ESpar\}\\mml@font@normal {}~has other downsides when compared with \\textsc \{ESamp\}\\mml@font@normal {}.
First, the memory consumption of \\textsc \{ESpar\}\\mml@font@normal {}~is O(mp) where p is a parameter, and is larger than ... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
"Ahmet Erdem Sariyuce",
"Srikanta Tirthapura"
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d1fd36bab477889b9e4ce9647d676658b19f9bd5 | subsection | 27 | 31 | Vertex Sampling (Algorithm | Overall, two algorithms take similar times to reach a 1\% error on all graphs we considered, in the range of 1.7-5 sec, with \\textsc \{ESpar\}\\mml@font@normal {}~achieving this accuracy faster than \\textsc \{ESamp\}\\mml@font@normal {}~with \\textsc \{Fast-eBFC\}\\mml@font@normal {}.
journal}{\\emph \{{\left(\@root ... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
"Ahmet Erdem Sariyuce",
"Srikanta Tirthapura"
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c9eea11b49fbc5e8765d20baeec264819dbe4fca | subsection | 28 | 31 | Vertex Sampling (Algorithm | To understand this, we begin with {lem:edge-spars,lem:clr-spars} which show expressions bounding the variances in terms of p_{1w}, p_{1e}, and p_{2v}, also summarized in {table:variance-sprs}. The difference in the variance between \textsc {ClrSpar}{}~ and \textsc {ESpar}{}~ boils down to (\mathbin {{}p^{-3}+p_{2v\bibi... | {
"cite_spans": []
} | 1801.00338 | Butterfly Counting in Bipartite Networks | [
"Seyed-Vahid Sanei-Mehri",
"Ahmet Erdem Sariyuce",
"Srikanta Tirthapura"
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d050f8779d646465e714ba3105051744d359a92a | subsection | 29 | 31 | Vertex Sampling (Algorithm | However, as shown in {figure:jrn-sprs-err-prob,figure:web-sprs-err-prob,figure:wiki-sprs-err-prob}, \textsc {ClrSpar}{}~{}requires a larger sampling probability to achieve a reasonable accuracy.
\bibitem {bad group22bad group22Lim15}
{author}{{\begin{}{31}
\end{}person}{Yongsub Lim} {and} {\bibliographystyle {plain}
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"Seyed-Vahid Sanei-Mehri",
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24bbbc024f472ec87dd9d510e5d13a7ab4674d92 | subsection | 30 | 31 | Vertex Sampling (Algorithm | Wu, personK. Yi, and
personZ. Li. year2016.
Counting Triangles in Large Graphs by Random
Sampling.
journalIEEE TKDE volume28,
number8 (year2016), pages2013–2026. | {
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d20927b79ed46bc09f21d009eafdf75c6957bf08 | abstract | 0 | 65 | Abstract | A new continuum-mechanical formulation is proposed which encompasses all
material processes within and surrounding an ice sheet. Using this formulation,
the balance of mass and free-surface relations for ice sheets are derived and
elaborated upon. The resulting three-dimensional mass-balance relation is then
integrated... | {
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951699edb71bc5fb22d6b75020ccf526314a8f45 | subsection | 1 | 65 | Introduction | Continuum-mechanical formulations which describe the dynamics of ice sheets and glaciers are complicated due to varied environmental interactions, including the atmosphere, lithosphere, sub- and supra-surface lakes, the oceans, etc; clearly-defined discontinuities within the ice-sheet interior; water transport within a... | {
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7129aba943c0b10b61adccb0c543e5d53f504852 | subsection | 2 | 65 | The multi-phase cryosphere | Ice sheets and glaciers may be represented as a system of differential multi-phase equations of disperse flow (cf. ).
These equations were first stated for binary ice and water mixtures by , which is a modification to concepts developed in .
The formulation developed here incorporates statements of vapor conservation, ... | {
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b73e4a2c1c22f12b708b4e09b623bebc2daef8d6 | subsection | 3 | 65 | The multi-phase cryosphere | Imagine within an ice-mixture mass density \rho = \rho (\underline{x},t) which is composed of some amount of solid (\mathrm {s}), liquid (\mathrm {\ell }), and vapor (\mathrm {v}) components, each moving relative to barycentric velocity \underline{u} = \underline{u}(\underline{x},t) defined by satisfying\rho ^{\pm } \u... | {
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b268bfadebb4c76ffb50731e1b0167f2da95d9a6 | subsection | 4 | 65 | The multi-phase cryosphere | For example, the atmosphere boundary will have \rho _{\mathrm {\ell }}^+ = \rho _{\mathrm {w}} in regions in contact with collected surface water, the basal surface may have a range of values \rho _{\mathrm {\ell }}^+ \in [0, \rho _{\mathrm {w}}] depending on the saturation of the sub-glacial aquifer, and \rho _{\mathr... | {
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94ca62300c49265dfe32abad6ea1457f463956c7 | subsection | 5 | 65 | The multi-phase cryosphere | The following proposition illustrates that source terms mixturedensitysource,surfacemixturedensitysource are in fact homogeneous:The total rate of change of the mass density \rho = \rho (\underline{x},t) and surface-mass density q = q(\underline{x},t) given by mixturedensitysource and surfacemixturedensitysource are re... | {
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8f495add8d8dd6e2e60d4a71d34a0fdf0e86ab49 | subsection | 6 | 65 | The multi-phase cryosphere | Finally, insertion of solidliquidreactionpair,solidvaporreactionpair,liquidvaporreactionpair into solidmassreaction,liquidmassreaction,vapormassreaction and taking the sum results in thmhomogeneoustotalmassreaction.The mass changes along surface encompassed by relations solidmassreaction,liquidmassreaction,vapormassrea... | {
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4f66b64ba7c44909224b13c2384da27fee14f111 | subsection | 7 | 65 | Surface-mass balance | The mechanisms which control the dynamics of ice sheets involve environmental interactions; hence an accurate description of the ice-sheet surface is essential.
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c42fd4bc59972c44d870c047daf7a08bc4d57532 | subsection | 8 | 65 | Component surface-mass balance | The transport of each solid (\mathrm {s}), liquid (\mathrm {\ell }), and vapor (\mathrm {v}) component masses within can only be governed by advectionThe velocity \underline{u} represents the motion of the component masses., while along the surface of the ice sheet , some fluctuation of each component mass will occur v... | {
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9c458dd7ee150bba9fd0c093c4e8462cd81d81e9 | subsection | 9 | 65 | Component surface-mass balance | Hence relations phasesurfacemassbalanceone,phasesurfacemassbalance evaluated from the ice-sheet (-) perspective with k = \mathrm {s} yields the Dirichlet component of the Navier boundary conditions given by\underline{u}_{\mathrm {s}}^- \cdot \underline{\hat{n}}^- = -\mathring{F}_{\mathrm {s}}^- = - \mathring{q}_{\mathr... | {
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bd2f50dba38cb34a5a35626cde87adadba114e2d | subsection | 10 | 65 | Mixture surface-mass balance | Using barycentric velocity \underline{u}^{\pm } from relation barycentricvelocity, the total mass jump over the surface is given by the sum of each component jump of componentmassjump:\left.{ \rho \left( \underline{u} - \underline{w} \right) } \right. = 0 && \text{on } ,where thmhomogeneoustotalmassreaction has been us... | {
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a6bf339b0bd07b3170627e8b1383ec52a02057a9 | subsection | 11 | 65 | Mixture surface-mass balance | These processes include phase transitions, densification of snow on the upper surface, basal-water flux due to internal-viscous heating, interaction with the basal-hydraulic system, accumulation of ice under a floating ice shelf due to super-cooled water that has frozen due to rising convection currents, etc.If the ext... | {
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cb8a355452daf9a36d3f515295f341d78763d19b | subsection | 12 | 65 | Ice-sheet surface-mass balance | Setting (\pm ) = (-) \iff (\mp ) = (+), removing all (-) superscripts from the ice variables, and collapsing the exterior (+) domain to the ice-sheet surface \rightarrow , relation surfaceflux is identical to the expressionSurface-mass balance relation surfacemassbalance is identical to Equations (5.19) and (5.29) of w... | {
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a0d802b624183529cd7bba4004886539ccfcd918 | subsection | 13 | 65 | Global/local mass conservation | Consider the ice-sheet domain (\underline{x},t) \in \mathbb {R}^3 with boundary = \partial (\underline{x},t) and outward-pointing-unit normal \underline{\hat{n}}.
Substitution of mass density \phi = \rho in Reynolds transport theorem (cf. thmreynoldstransport) yields\frac{\mathrm {d}^{} }{\mathrm {d} t^{} } \int _{} \r... | {
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ed948bf7b46f983cc1b83a4bbbf681dbf990f056 | subsection | 14 | 65 | Local incompressibility | Consider within an arbitrarily fixed ice-sheet material element ^{\mathrm {e}}(\underline{x}) \subset with boundary \partial ^{\mathrm {e}} = ^{\mathrm {e}}.
The constant volume of ^{\mathrm {e}} demands that density fluctuations caused by melting or freezing—induced by changes in pressure and heat flux—present a mass ... | {
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ca6951665637eb857d7b0d0263ff457ca52da457 | subsection | 15 | 65 | Local incompressibility | Therefore, provided that mass transport is conserved such that \dot{m} = 0, relation mixturemassbalance is reduced toRelation incompressibleconservationofmass is identical to Equations (2.11)_1 of and (3.60) of with \rho _v \equiv 0.\nabla \cdot \underline{u} &= 0 &&\text{ in } ;the well-known incompressibility constra... | {
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d496b8200dba3850a6fedf8402addfaf194dd3b6 | subsection | 16 | 65 | Free-surface equation | The results of this section are identical to that of .
For reasons stated in the introduction, and for clarity of the analysis to follow, the fundamental free-surface relations are revisited.
To this end, assume that the surface of the ice-sheet body can be stated in implicit form in the sense of thmimplicitsurface.
Th... | {
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076ad9f945dfee987f1f18d63e0e47d0b09c4177 | subsection | 17 | 65 | Free-surface equation | It is common practice to first solve for \underline{u} at time t from momentum conservation, then solve intermediatekinematicsurface for an updated surface F(\underline{x}, t + \Delta t) some interval of time \Delta t from t.Decomposition of the surface into = _{\mathrm {S}} \cup _{\mathrm {B}} with _{\mathrm {S}} \cap... | {
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485634cec643c7ca269ff3411a3cb6910b8e97d1 | subsection | 18 | 65 | Free-surface equation | Therefore, for an inclined surface,
\Delta S = \mathring{S}\Delta t ( \hat{\underline{k}} \cdot \underline{\hat{n}})^{-1} = \mathring{S}\Delta t \Vert \hat{\underline{k}} - \nabla S \Vert ,
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\partial... | {
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ef9f6923015c3d0b9a4eb81dcb4bf4e3ee8f05fd | subsection | 19 | 65 | Mass balance in | Similar to secfreesurface, the results of this section are identical to that of .
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bd196d4c57bbf6fb486f0fab889a057341b02368 | subsection | 20 | 65 | Mass balance in | Integrating volume-conservation relation incompressibleconservationofmass vertically produces\int _B^S \nabla \cdot \underline{u} \ \mathrm {d}z
= & \int _B^S \left( \frac{\partial ^{} u_x}{\partial x^{} } + \frac{\partial ^{} u_y}{\partial y^{} } + \frac{\partial ^{} u_z}{\partial z^{} } \right) \ \mathrm {d}z = 0.The... | {
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caf45b4e099554337f69173319b22a091c450a91 | subsection | 21 | 65 | Mass balance in | In addition, the surface-normal-vector magnitudes (cf. defnormalvector)\Vert \nabla F_{\mathrm {S}} \Vert = \Vert \hat{\underline{k}} - \nabla S \Vert \ge 1
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b11d0db9bea2b0dc8ff7b3c3deeee151361426e9 | subsection | 22 | 65 | Error analysis | If the upper and lower surfaces are relatively flat, it has been previously assumed that (cf. rmknormalkinematicforcing, rmksmbcoefficients, and rmkmisconception)\Vert \hat{\underline{k}} - \nabla S \Vert \equiv 1,
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08c1c17c119a496b5e1e9fc178e2945a668b4f13 | subsection | 23 | 65 | Error analysis | However, the flanks of Jakobshavn's trench possess a abnormally high surface-gradient magnitude (figjakobgradb); coupled with the fact that this area is characterized by very high magnitudes of basal velocity and basal-mass balance \mathring{B}, assumptions commonassumptions will induce a significant error in the forci... | {
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44e9722e26b527fd82c675773925c8d1b9eab3fb | subsection | 24 | 65 | Error analysis | In addition, the ice sheet flow will increase in magnitude as the surface gradient magnitude increases (cf. , ); therefore, the imposition of assumptions commonassumptions will generate a non-physical augmentation of velocity that is of greatest magnitude near regions of high surface slope.
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4591006ea824b56ca7bbf0e6aa5f66f3047e6c2e | subsection | 25 | 65 | Analytic solution | An analytic solution satisfying incompressibility relation incompressibleconservationofmass and free-surface equations upperfreesurface,lowerfreesurface provides the ability to verify and thereby guarantee the correct implementation of any numerical model associated with ice-sheet mass conservation.
The verification of... | {
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eea65ab2a9afed3a455732698739a6ac97a530e4 | subsection | 26 | 65 | Analytic solution | That is, the linearly-interpolated analytic vertical component of velocity is given byu_z^{\mathrm {a}}(\underline{x},t) = \xi _{\mathrm {S}} u_{z \mathrm {S}}^{\mathrm {a}} + \xi _{\mathrm {B}} u_{z \mathrm {B}}^{\mathrm {a}},where\xi _{\mathrm {B}}(\underline{x},t) = \frac{S(x,y,t) - z}{H(x,y,t)},
\hspace{28.45274pt}... | {
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4db8333274713f1c3d561c76040792fefa7b733a | subsection | 27 | 65 | Analytic solution | Thus u_z^{\mathrm {a}} defined by analyticzvelocity is fully determined once S, B, u_x^{\mathrm {a}} and u_y^{\mathrm {a}} have been specified.For simplicity, the x component of velocity is chosen to beu_x^{\mathrm {a}}(\underline{x},t) &= \left( u_{x \mathrm {S}}^{\mathrm {a}} - u_{x \mathrm {B}}^{\mathrm {a}} \right)... | {
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3944c6ad6b7039b334c1c1d8c817798aaa33beeb | subsection | 28 | 65 | Analytic solution | Hence this parameter may be used to verify the numerical implementation of the Dirichlet boundary condition \mathring{B}= - \underline{u} \cdot \underline{\hat{n}} over basal surfaces (cf. rmkimpenetrablesurfaceremark).The final y component of velocity u_y^{\mathrm {a}} must satisfy incompressibility relation incompres... | {
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d6730c7be94dd7b9d189acbc3308b23010eb60eb | subsection | 29 | 65 | Analytic solution | In addition, as the coefficients A, G, and C do not depend on u_y^{\mathrm {a}}, equation analyticyvelocityproblem is linear; the authors of and incorrectly describe their analogous relations as quasi linear.
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3a92f8a3638e1739fa0867eca0faea7509b35e4a | subsection | 30 | 65 | Analytic solution | In this case, application of the chain rule (cf. thmonevariablechainrule) produces\frac{\mathrm {d}^{} u_y^{\mathrm {a}}}{\mathrm {d} s^{} } = \frac{\partial ^{} u_y^{\mathrm {a}}}{\partial y^{} } \frac{\mathrm {d}^{} y}{\mathrm {d} s^{} } + \frac{\partial ^{} u_y^{\mathrm {a}}}{\partial z^{} } \frac{\mathrm {d}^{} z}{... | {
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89cba9144dde129f6364688a8c9f54155a9e9fd8 | subsection | 31 | 65 | Analytic solution | This function may be arbitrarily specified in terms of the coordinates \phi _1 and \phi _2; for simplicity, let
\vartheta (\phi _1, \phi _2) \equiv \lbrace \phi _1(y,z) = 0,\ \phi _2(y,z) = 0\rbrace \iff z_0 = 0,\ u_{y0} = 0.
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0f679d98998fdbb0d6d332acdd1722b00b06b768 | subsection | 32 | 65 | Analytic solution | The fourth and fifth integrals of analyticyvelocity given by\mathcal {I}_{\mathrm {S}} = \frac{1}{H} \int _y \Vert \hat{\underline{k}} - \nabla S \Vert \mathring{S}\ \mathrm {d}y
\hspace{28.45274pt}
\text{and}
\hspace{28.45274pt}
\mathcal {I}_{\mathrm {B}} = \frac{1}{H} \int _y \Vert \nabla B - \hat{\underline{k}} \Ver... | {
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da5e3f49081504932ee7dada46789008d345eff6 | subsection | 33 | 65 | Example calculation | A specific realization of the solution derived in secr3analyticsolution is hereby generated (figr3) over the ice-sheet domain = [0,\ell ] \times [0,\ell ] \times [B,S] \subset \mathbb {R}^3 with upper and lower surfacesThese surfaces were chosen independent from y as suggested by rmkellipticintegral.S(x) = \frac{1}{10}... | {
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1f63b5c6be04b481a09a4d0f491dfb53499e4f98 | subsection | 34 | 65 | Example calculation | Finally, the upper and lower surface-mass balance terms were respectively chosen to be (figr3smb,figr3bmb)\mathring{S}= \sin \left( \frac{4 \pi }{\ell } x \right) \sin \left( \frac{4 \pi }{\ell } x \right)
\hspace{22.76219pt}
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\hspace{22.76219pt}
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b8200f0f09a9b491ea3c2a8500c2946e5c7a9967 | subsection | 35 | 65 | Future work | The concepts of mass balance derived here provide the basis of a new formulation for energy and momentum conservation for ice sheets.
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bfbfdebf987105a96e20067c54e10e1ec9bcfbe8 | subsection | 36 | 65 | Mathematical background | [Figure: NO_CAPTION]Behavior of the Taylor coefficients as h decreases. It is common to refer to the truncated Taylor series which does not include the h^n terms as the \mathrm {n}th-order-Taylor-series approximation which is clearly only accurate for h \ll 1.This section introduces the relevant mathematical background... | {
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4b15917b08fa8db14bed2aed283fa8652dd44fd9 | subsection | 37 | 65 | Mathematical background | The basis from which we begin is the following fundamental theorem presented by :[Taylor's theorem]
Any real-valued function f(x) \in \mathbb {R} with x \in \mathbb {R} that is infinitely differentiable about a point x+h with distance h>0 may be expressed as the infinite Taylor seriesf(x+h) &= f(x) + f^{\prime }(x) h ... | {
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69819e4a58da372e0f50115ddcfb124908bff9b2 | subsection | 38 | 65 | Mathematical background | Similar to the line of reasoning resulting in defderivative, taking the limit of small \Vert \underline{h} \Vert suggests the following definition:[Gradient]
The gradient of a real-valued function f(\underline{x}) \in \mathbb {R} with respect to its vector of coordinates \underline{x} = [x_1\ x_2\ \cdots \ x_n]\mathbb... | {
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0e790dc729c20f16950e7b820f624b8b96ccc4b3 | subsection | 39 | 65 | Mathematical background | Then the derivative of f with respect to t is\frac{\partial ^{} f}{\partial t^{} } = \frac{\partial ^{} f}{\partial x_1^{} }\frac{\partial ^{} x_1}{\partial t^{} } + \frac{\partial ^{} f}{\partial x_2^{} }\frac{\partial ^{} x_2}{\partial t^{} } + \cdots + \frac{\partial ^{} f}{\partial x_n^{} }\frac{\partial ^{} x_n}{\... | {
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e4fb83030e185b426760d2031915a5dae5c49dc7 | subsection | 40 | 65 | Mathematical background | Using the fact that the value of \underline{h} = \underline{h}(\underline{x}, t) will be identical at each instant t, the derivative of the function F with respect to t is given simply byThe overhead-dot notation (\ \dot{ }\ ) was first used by to specifically denote differentiation with respect to time. Other rates of... | {
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fb1569a5b70d490ef3d195a77f23f00dd7e3be0b | subsection | 41 | 65 | Mathematical background | Finally, the literature commonly refers to the second term u f as the \emph {advective} component due to the fact that it is responsible for the transport of the quantity f at speed u.\footnote {The process of advection can be better understood by consideration of fundamental-conservation equation {def_fundamental_cons... | {
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8507b7260e22401bebf2e5913881700938376a03 | subsection | 42 | 65 | Mathematical background | Then F(\underline{s}(t)) = 0 for all t and application of the chain rule (cf. thmonevariablechainrule) yields\frac{\mathrm {d}^{} F}{\mathrm {d} t^{} } = &\frac{\partial ^{} F}{\partial x^{} } \frac{\mathrm {d}^{} x}{\mathrm {d} t^{} } + \frac{\partial ^{} F}{\partial y^{} } \frac{\mathrm {d}^{} y}{\mathrm {d} t^{} } +... | {
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2ba40bad376c3e322c26c2fb8258e62172eb9664 | subsection | 43 | 65 | Mathematical background | The outward-pointing unit-normal vectors over these surfaces are respectively\underline{\hat{n}}_{\mathrm {S}} = \frac{ \nabla F_{\mathrm {S}} }{\Vert \nabla F_{\mathrm {S}} \Vert }
\hspace{28.45274pt} \text{and} \hspace{28.45274pt}
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1be52830ae56060e2266da20baa15d4520041055 | subsection | 44 | 65 | Mathematical background | Hence the area of A_i is given by\Vert \underline{u}_i \times \underline{v}_i \Vert = \left(1 + \left( \frac{\partial ^{} S_i}{\partial x^{} } \right)^2 + \left( \frac{\partial ^{} S_i}{\partial y^{} } \right)^2 \right)^{\frac{1}{2}} \Delta x_i \Delta y_i.Using the normal-vector magnitude of defnormalvector, summing ov... | {
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63118a0c6ddbf8d907b290662131cdc7b7abe64d | subsection | 45 | 65 | Mathematical background | That is,\frac{\partial ^{} \mathcal {I}}{\partial x^{} }
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