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[MATH] in which a temporal accumulation of random growth rate [MATH] contributes to make the inequality even for the uniform randomness at each time step. Because the distribution follows a log-normal from the central limit theory for |
[MATH] Similar distributions have been obtained in a simple model of random copying among individuals in the evolution for studying cultural change |
The distributions consist of many uncommon variants and a very few common variants, where cultural variants means first names, journal citations, decorative motifs on archaeological pottery, and patent citations in U.S. Thus, random copying may induce a key mechanism to self-organize a complex structure depending on a ... |
It suggests that complex network structures can be random process In this paper, we consider a self-organized design of efficient and robust networks by biologically inspired copying. In particular, we focus on the robust network structure with positive degree-degree correlations |
and an incrementally growing mechanism. The self-propagation in maintaining the robust structure is particularly important for the scalability of network system without degrading the performance. |
The organization of this paper is as follows. In Sec. II , we briefly review related network models to ours. In Sec. III we propose a basic model for understanding the fundamental mechanism of network self-organization from a viewpoint in complex network science |
which aims to elucidate the universal properties and the generation rule of networks. We show the important properties of our model for the robustness of connectivity and the emergent functions. In Sec. IV , we consider an incrementally growing mechanism, and investigate the strong robustness and the efficiency for pat... |
II Related Works in Complex Network Science It is a reasonable assumption that a biological network grows according to a simple mechanism based on random copying which differs from the preferential attachment |
by selecting large degree nodes in selfishness. Thus, in the duplication-divergence (D-D) model a new node is added at each time step, and duplicately creates links to connected neighbor nodes of a randomly chosen node. Some links in the duplication are deleted with a probability [MATH] Without using degrees for the se... |
and by D-D with weak deletion of small [MATH] have a singular property called as non-self-averaging In the pure duplication bipartite graphs [MATH] in any combinations of positive integers [MATH] and [MATH] are generated with equiprobability, the degrees of [MATH] and [MATH] distribute uniformly. The statistical quanti... |
of links in a sample of the networks have no meaning to characterize the topological structure since the range between the minimum [MATH] at [MATH] |
and the maximum [MATH] at [MATH] diverges for a large size [MATH] In addition, the attack vulnerability remains in the D-D model because of the SF-like network. |
On the other hand, the optimal structure against the targeted attacks to hub nodes has been shown numerically and analytically in a SF network. It is referred to onion-like structure (see Fig. from the character of connectivity with positive degree-degree correlations, which correspond to a natural tendency of homophil... |
Thus, the onion-like structure is applicable to a network with any degree distribution. For example, after the rewirings, the attack vulnerability is decreased for a power-law degree distribution in a SF network preferential attachment or other methods. However, |
the entire rewiring process discards already existing links, it is not effective utilization when a network is growing in a realistic situation Moreover, in spite of the improvement of robustness, the positive degree-degree correlations in a SF network tend to induce longer path lengths |
, which are undesirable for the efficient communication or transportation. We should not persist SF networks, and study other candidates for future design of networks, especially in considering self-organization mechanisms. |
III Basic Properties In this section, we consider a self-organization of onion-like network, and show the tolerant and topological properties of robustness, degree distribution, and degree-degree correlation. To study the connectivity in a network structure independently of applications is important for clarifying the ... |
III-A Network model We propose a modification of the D-D model without mutation. Mutations by connecting randomly chosen two nodes are unnecessary for generating an onion-like network. The basic processes of network construction by using only local information are as follows (see Fig. ). |
Step 0: Set an initial configuration (e.g. connected two nodes). Step 1: At each time step [MATH] a new node is added. The new node [MATH] connects to a randomly chosen node and to the neighbor nodes [MATH] |
with a probability [MATH] [EQUATION] where [MATH] is a rate of link deletion, [MATH] is a parameter, and [MATH] and [MATH] denote the degrees of nodes [MATH] and [MATH] |
Step 2: The above processes are repeated up to a given size [MATH] in prohibiting self-loops at a node and duplicate connections between two node. |
In Step 1, [MATH] is temporary set as [MATH] the degree of the chosen node, since the degree of new node is unknown in advance because of the stochastic process. When degrees [MATH] and [MATH] |
are close, the two nodes [MATH] and [MATH] are connected with high probability. The case of [MATH] without the correlation effect corresponds to the conventional D-D model except the mutual link between new node and chosen node. We set [MATH] |
Figure shows that our proposed networks satisfy a good property of the self-averaging unlike the conventional D-D model: the statistical deviation |
[MATH] for the total number [MATH] of links converges to zero for a large size [MATH] Here, [MATH] denotes the statistical mean (expectation). We remark a reason of the self-averaging by that a sequence of complete graphs is generated in our case of pure duplication of [MATH] instead of bipartite graphs in the conventi... |
Therefore, with a small change from the conventional D-D model by adding the mutual links, the tendency of dense connections (whose extreme case is a complete graph) induces a core of connected high degree nodes. In addition, since there is an effect of preferential attachment to the random neighbors |
, large degree nodes [MATH] and [MATH] tend to be connected together when the chosen node has a large degree. The positive degree-degree correlations is suitable to improve the robustness, especially for the malicious attacks |
However, such correlations between low degree nodes are weak in the tree-like structure as shown in the top of Fig. Since a low degree node dangles from a higher degree node, the dangling part is easy to be disconnected by node removals. We remark that the majority is low degree nodes. Thus, we consider the addition of... |
in Step 3. Step 3 : After Step 2, some shortcut links between randomly chosen nodes [MATH] and [MATH] are added with the probability of Eq.( ). The number of shortcut links are given by [MATH] where [MATH] is an adding rate of shortcut and [MATH] is the total number of links in the original tree-like network generated ... |
The bottom of Fig. shows an onion-like structure by adding shortcut links after Step 2. In the next subsection, as a basic property, we will separately discuss the effects of copyings and adding shortcuts on the robustness in order to make them clearly. |
We emphasize the two important functions. First, a new node act as a local proxy of the chosen node and becomes another access point for the neighbor nodes in an analogy of distributed computer communication systems. Second, the complementary added shortcuts improve the robustness It has already been shown that adding ... |
and also in the numerical simulations for many networks The robustness is further improved due to the positive degree-degree correlations in the onion-like network as shown later, however the case of [MATH] as the uniform random selections does not give the nearly optimal robustness. |
In addition, the proposed network has an efficiency without maintaining huge connections. Figure shows that the degree distributions are approximated by exponential distributions. Therefore, huge degree nodes do not appear in the network, it is suitable for avoiding both the attack vulnerability |
and the concentration of linking cost or traffic load Note that the proposed network does not belong to a SF network, since the degree distribution is not a power-law but an exponential. |
III-B Measures of robustness and degree-degree correlations We consider the malicious attacks, in which nodes are removed in decreasing order of the current degrees through the recalculations |
This type of attacks is a more serious problem setting from an assumption of circumspect terrorism than the attacks without recalculations. Attackers tend to aim at large degree nodes, since the removals give considerable damage to maintaining paths by many disconnections emanated from the removed nodes. We compare the... |
with a probability [EQUATION] the connections of randomly selected nodes [MATH] and [MATH] are repeated. Here, [MATH] and [MATH] denote the rank for the degrees of nodes [MATH] and [MATH] A remaining small part of unpaired stubs is remedied by the reshuffle procedure. In the stub-connection process, the initial configu... |
[MATH] has [MATH] free links (whose linked nodes are undetermined) assigned from a given degree distribution [MATH] We have two measures for investigating the robustness and the degree-degree correlations. The robustness is measured by the following index |
[EQUATION] where [MATH] denotes the number of nodes in the giant component (GC: largest connected cluster) after removing [MATH] nodes by the malicious attacks. |
[MATH] is monotonically decreased for increasing [MATH] from an initial configuration of [MATH] The range of [MATH] is [MATH] , where [MATH] corresponds to a completely disconnected network consisting of isolated nodes, and [MATH] corresponds to the most robust network. |
The degree-degree correlation is measured by the assortativity [EQUATION] where [MATH] [MATH] [MATH] [MATH] [MATH] denotes the [MATH] [MATH] element of the adjacency matrix. The right-hand side of Eq. ( ) is a suitable scheme for the numerical calculation of [MATH] |
than the original definition The range of [MATH] is [MATH] as the Pearson correlation coefficient of the degree. Nodes with similar degrees tend to be connected as |
[MATH] is larger, while nodes with different degrees tend to be connected as [MATH] is smaller (but [MATH] is larger). Figure shows a scatter plot of robustness index [MATH] versus assortativity [MATH] The open marks correspond to the proposed networks, and the filled marks correspond to the rewired versions at the opt... |
We set [MATH] for the added versions with shortcut links for rates [MATH] and [MATH] The values of [MATH] become larger as [MATH] is smaller in the tree-like networks with more links (denoted by open triangles and diamond marks). While the robustness is improved by adding shortcut links (denoted by open square and circ... |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] and saturated around the nearly optimal for the rewired versions (denoted by filled marks). The dashed arrows show the improvement for the robustness, and the almost flat ones mean that the proposed networks with shortcuts have the near... |
Not all assortative networks have onion structure but all onion networks are assortative IV Growing self-organization We further consider an incrementally growing onion-like networks by simultaneous processes of copyings and adding shortcuts. Because the tree-like network leaves its robustness weak in the growth except... |
Step 2’: In Step 2, at every interval [MATH] shortcut links are added between randomly chosen nodes [MATH] and [MATH] according to the probability of Eq.( ). This process is repeated up to [MATH] links. Here, [MATH] denotes the total number of links in the network at that time |
[MATH] In this section, we numerically show the strong robustness against both attacks and failures and the efficiency for path lengths in the incrementally growing onion-like networks. Not only the emergent structure in distributed manner but also the scalability without performance degradation are important for the s... |
IV-A Robustness in the incrementally growing networks In order to be [MATH] at [MATH] with [MATH] we chose a combination of parameters: |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] and the corresponding [MATH] in the tree-like networks. Since the robustness of connectivity depends on a degree distribution but increases as the average degree [MATH] is larger with more links in general, a same level of connection density must be set to compare... |
denotes the initial size ( [MATH] when the initial configuration is a connected two nodes). Figure (a) shows the relative size [MATH] of nodes belonging to the GC versus the fraction [MATH] of removed nodes by the malicious attacks. From the tree-like (denoted by cyan dashed line) to the onion-like (denoted by other li... |
for the onion-like networks but very small [MATH] for the tree-like networks as shown in Fig. (b) and the inset. At the peak, the GC breaks off and is divided into small clusters. Note that the critical fraction [MATH] is around [MATH] |
for the SF network models and the Internet data at the level of autonomous system Figure (a) shows that the values of [MATH] is sustained without a considerable drop against random node removals, except the case of [MATH] (denoted by cyan dashed line). Note that the virtual line of angle [MATH] |
is the most robust case of [MATH] in the complete graph with the maximum [MATH] : wasteful links. The critical fraction [MATH] is about [MATH] |
for the onion-like networks but [MATH] for the tree-like networks as shown in Fig. (b). Remember that the robustness index [MATH] defined in Eq.( indicates the area under a line of [MATH] Thus, higher robustness (values of [MATH] ) in the growing onion-like networks is obtained than that in the tree-like networks for [... |
are summarized in Table II As the deletion rate [MATH] is larger, [MATH] is smaller, while [MATH] is almost constant in both cases of the attacks and the failures. |
We also study a spatially growing method. We consider the initial configuration of connected two nodes located at the center with the link length [MATH] in a [MATH] square. At each time step in Step 1, from the center of a randomly chosen node, a new node is set on the radius of random number between [MATH] |
and [MATH] with any direction as similar to the basic idea of We set [MATH] and [MATH] As shown in the top of Fig. the growing networks spread out diffusively on the space according to the time course. Figure 10 shows the relation of |
[MATH] and [MATH] through the growth. In the incrementally growing onion-like networks for the values of [MATH] and [MATH] the marks of green plus, red cross, blue open square, and magenta open circle on each line correspond to the size [MATH] from left to right. Therefore, both [MATH] and [MATH] increase with the time... |
[MATH] slightly decreases in the tree-like networks denoted by the marks of cyan triangle and black filled circle. The directions of lines are from top to bottom for |
[MATH] in the time course. The long upper jump of a gray dashed line is due to become onion-like networks finally at [MATH] by adding many shortcut links with the rate [MATH] |
in Step 3. Through the growth, the assortativity [MATH] is almost constant around [MATH] except it increases (from [MATH] to [MATH] in the case of [MATH] [MATH] |
IV-B Efficiency for path lengths The efficiency for communication or transportation is measured by the average path lengths on the shortest paths over the network. The length is usually counted by the minimum number of hops between two nodes. Because a path which passes through as few mediator nodes as possible gives l... |
the frequency of passing through a node or a link on the paths between all pairs of nodes. The frequency can be weighted by an inhomogeneously distributed (communication or transportation) requests, e.g. proportional to the products of populations around source and destination nodes on a space |
in more realistic situation. Figure 11 shows that the distributions of path lengths in the networks at a same connection density level of |
[MATH] The widths are narrower around 5 hops in the growing onion-like networks denoted by green dashed, red solid, blue dashed, and magenta dotted lines for [MATH] [MATH] [MATH] and [MATH] |
than that in the tree-like networks denoted by cyan chain line for [MATH] The path lengths are short around 5 hops in average and at most about 10 hops for [MATH] Thus, the growing onion-like networks are efficient with short paths. |
In more detail for the emergence of SW property it is important whether the average path length follows [MATH] for the size [MATH] under a constant connection density. Inset of Fig. 12 shows the SW property in the tree-like networks denoted by cyan and orange chain lines for [MATH] and [MATH] Moreover, as shown in Fig.... |
(longer path lengths) in [MATH] denoted by magenta dotted line for [MATH] than the case of tree-like networks denoted by cyan and orange chain lines for [MATH] and [MATH] We should remark that the average degree [MATH] |
is not constant within a connection density but increasing in the growing onion-like networks. Moreover, [MATH] is smaller than that in the tree-like networks for [MATH] as shown in Fig. 10 Smaller [MATH] means fewer total links, and the network construction is less expensive in lower cost especially in an early stage ... |
Another important measure is the path distance, which is defined by the sum of Euclidean distances for the links on the shortest path (counted by the minimum number of hops) between two nodes. This measure is corresponding to the load of physical movements for (communication or transportation) flows on a space, instead... |
Figure 13 shows the distribution of path distances [MATH] , in which the majorities are short around the size [MATH] of outer square. The case of [MATH] as the tree-like network denoted by cyan chain line has longer distances than other cases, since the distribution is shift to the right. We also confirm the SW propert... |
in the onion-like networks are below the cyan and orange chain lines for [MATH] and [MATH] in the tree-like networks, except the magenta dotted line for [MATH] Thus, the onion-like networks has shorter path distances than the tree-like networks within a same connection density level, although the advantage in the Eucli... |
12 and 14 ). Note that we have similar results for the path length [MATH] and distance [MATH] , when we chose the shortest distance path instead of the above shortest path defined by the minimum number of hops between two nodes. The difference of shortest (Euclidean) distance path and shortest (minimum hops) path is ve... |
Discussions for applications In this section, we will discuss the possibilities or issues in our proposed networks for applications of communication or transportation systems. There exist flows which represent movements of one entity relative to another on a network. Thus, we categorize the dynamics in networks into th... |
Type 1. Dynamics of network configuration itself Type 2. Dynamics of information flows, rumor spreading, opinion formation, synchronization, or logistics on a network |
Through this paper, we study the Type 1 of dynamics in order to aim a fundamental mechanism for the self-organized design of efficient and robust networks. Temporal and/or fixed (corresponding to wireless and/or wired) connections are possible depending on the time-scale for changing the connection structure in a netwo... |
are necessary. There are many methods for the efficient operations that are independent (generally applicable) or dependent on a special network structure. The detail discussions are strongly related to device technologies, users, and situations of utilization, they are beyond our current scope. |
Even when we focus on the Type 1 of dynamics, there are two kinds of interactions among individuals (nodes) and with environment |
for the operation and control in a self-organized system. In particular, we consider changes of environment and the adaptivity in a network. Here, adaptivity is defined as |
the capability to work in different or changing environment without intervention and configuration by an administrator The levels of adaptation are distinguished as shown in Table III |
At the Level 1, a compensatory growth for removed parts by attacks or failures is related to an adaptivity with healing function in the incrementally growing onion-like networks because of the potential with strong robustness. The function of proxy at a new node locally contributes to make different access paths throug... |
As a base structure for DTN, it is obviously better to have many short paths and to maintain tolerant path connectivity for temporal changes of nodes or links. Our growing onion-like network becomes one of the candidates of DTN. |
At the Level 2, in our growing network, it will be useful to regulate the parameters of [MATH] [MATH] , and [MATH] in order to repair the damaged parts and to recover the performance for communication or transportation according to a limited resource and the state of network system monitored at a time. The scheduling i... |
Apart from the basic property, only random failures and malicious attacks by removing nodes in decreasing order of their current degrees are insufficient to investigate the robustness in a realistic network. Natural disasters such as earthquake or flood and unremitting armed conflicts with aerial bombing give rise to s... |
VI Conclusion We have proposed a self-organized design method for generating efficient and robust networks. It is based on biologically inspired copyings and the complementary adding shortcut links to enhance the robustness. In particular, tanking into account positive degree-degree correlations, we have considered inc... |
non-singular self-averaging property unlike the conventional D-D model [MATH] to ensure statistically meaningful convergence for a large size [MATH] in samples |
growing with the compensatory function of local proxy as another access point exponential degree distribution without huge hubs [MATH] to avoid the concentration of linking cost and traffic load at a few nodes |
strong tolerance of connectivity against failures and attacks equivalent to the nearly optimal in the rewired version under a same degree distribution |
efficiency with short path lengths and distances (counted by the number of hops and the Euclidean distances, respectively, on the shortest path) for communication or transportation, in addition the path length is superior to the [MATH] SW property |
in a sense Moreover, we have discussed the possibilities and issues for applications in wireless and/or wired communication or transportation systems, and particularly mentioned the adaptivity to heal over and to recover the performance of networking for a change of environment in such disasters or battlefields on a ge... |
Thus, the obtained results will be useful for temporal evolution of a resilient network system. Other capabilities and realistic strategies for self-diagnosis, self-protection, self-healing, self-repair, and self-optimization |
are further issues. We will study effectively applicable methods to self-organize networks which keep high robustness and efficient paths for communication or transportation in both normal and emergent situations. Acknowledgment This research is supported in part by a Grant-in-Aid for Scientific Research in Japan, No. ... |
# Source: arxiv 1501.03632 # Title: Fixation probability of rare nonmutator and evolution of mutation rates # Sections: all # Downloaded: 2026-03-03T01:59:12.863481+00:00 |
Fixation probability of rare nonmutator and evolution of mutation rates Abstract Although mutations drive the evolutionary process, the rates at which the mutations occur are themselves subject to evolutionary forces. Our purpose here is to understand the role of selection and random genetic drift in the evolution of m... |
and we address this question in asexual populations at mutation-selection equilibrium neglecting selective sweeps. Using a multitype branching process, we calculate the fixation probability of a rare nonmutator in a large asexual population of mutators, and find that a nonmutator is more likely to fix when the deleteri... |
Using these results for the fixation probability and a drift-barrier argument, we find a novel relationship between the mutation rates and the population size. We also discuss the time to fix the nonmutator in an adapted population of asexual mutators, and compare our results with experiments. |
KEY WORDS: mutation rates, branching process, fixation probability, fixation time Running Title : Evolution of mutation rates Contact Information (for all authors) |
Ananthu James postal address : Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560064, India |
work telephone number : +91-80-22082967 E-mail : ananthujms@jncasr.ac.in Kavita Jain postal address : Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560064, India |
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