5,073 research outputs found
On the density of critical graphs with no large cliques
A graph is \textit{-critical} if and every proper
subgraph of is -colorable, and if is a list-assignment for
, then is \textit{-critical} if is not -colorable but every
proper induced subgraph of is. In 2014, Kostochka and Yancey proved a lower
bound on the average degree of an -vertex -critical graph tending to for large that is tight for infinitely many values of ,
and they asked how their bound may be improved for graphs not containing a
large clique. Answering this question, we prove that for , if is sufficiently large and is a -free -critical graph where and is a
list-assignment for such that for all , then
the average degree of is at least . This result implies that for some , for every
graph satisfying where is the size of the largest clique
in and is the maximum average degree of , the
list-chromatic number of is at most .Comment: 26 page
Directed network modules
A search technique locating network modules, i.e., internally densely
connected groups of nodes in directed networks is introduced by extending the
Clique Percolation Method originally proposed for undirected networks. After
giving a suitable definition for directed modules we investigate their
percolation transition in the Erdos-Renyi graph both analytically and
numerically. We also analyse four real-world directed networks, including
Google's own webpages, an email network, a word association graph and the
transcriptional regulatory network of the yeast Saccharomyces cerevisiae. The
obtained directed modules are validated by additional information available for
the nodes. We find that directed modules of real-world graphs inherently
overlap and the investigated networks can be classified into two major groups
in terms of the overlaps between the modules. Accordingly, in the
word-association network and among Google's webpages the overlaps are likely to
contain in-hubs, whereas the modules in the email and transcriptional
regulatory networks tend to overlap via out-hubs.Comment: 21 pages, 10 figures, version 2: added two paragaph
Where Graph Topology Matters: The Robust Subgraph Problem
Robustness is a critical measure of the resilience of large networked
systems, such as transportation and communication networks. Most prior works
focus on the global robustness of a given graph at large, e.g., by measuring
its overall vulnerability to external attacks or random failures. In this
paper, we turn attention to local robustness and pose a novel problem in the
lines of subgraph mining: given a large graph, how can we find its most robust
local subgraph (RLS)?
We define a robust subgraph as a subset of nodes with high communicability
among them, and formulate the RLS-PROBLEM of finding a subgraph of given size
with maximum robustness in the host graph. Our formulation is related to the
recently proposed general framework for the densest subgraph problem, however
differs from it substantially in that besides the number of edges in the
subgraph, robustness also concerns with the placement of edges, i.e., the
subgraph topology. We show that the RLS-PROBLEM is NP-hard and propose two
heuristic algorithms based on top-down and bottom-up search strategies.
Further, we present modifications of our algorithms to handle three practical
variants of the RLS-PROBLEM. Experiments on synthetic and real-world graphs
demonstrate that we find subgraphs with larger robustness than the densest
subgraphs even at lower densities, suggesting that the existing approaches are
not suitable for the new problem setting.Comment: 13 pages, 10 Figures, 3 Tables, to appear at SDM 2015 (9 pages only
ĂlĆlĂ©nyek kollektĂv viselkedĂ©sĂ©nek statisztikus fizikĂĄja = Statistical physics of the collective behaviour of organisms
Experiments: We have carried out quantitative experiments on the collective motion of cells as a function of their density. A sharp transition could be observed from the random motility in sparse cultures to the flocking of dense islands of cells. Using ultra light GPS devices developed by us, we have determined the existing hierarchical relations within a flock of 10 homing pigeons. Modelling: From the simulations of our new model of flocking we concluded that the information exchange between particles was maximal at the critical point, in which the interplay of such factors as the level of noise, the tendency to follow the direction and the acceleration of others results in large fluctuations. Analysis: We have proposed a novel link-density based approach to finding overlapping communities in large networks. The algorithm used for the implementation of this technique is very efficient for most real networks, and provides full statistics quickly. Correspondingly, we have developed a by now popular, user-friendly, freely downloadable software for finding overlapping communities. Extending our method to the time-dependent regime, we found that large groups in evolving networks persist for longer if they are capable of dynamically altering their membership, thus, an ability to change the group composition results in better adaptability. We also showed that knowledge of the time commitment of members to a given community can be used for estimating the community's lifetime. Experiments: We have carried out quantitative experiments on the collective motion of cells as a function of their density. A sharp transition could be observed from the random motility in sparse cultures to the flocking of dense islands of cells. Using ultra light GPS devices developed by us, we have determined the existing hierarchical relations within a flock of 10 homing pigeons. Modelling: From the simulations of our new model of flocking we concluded that the information exchange between particles was maximal at the critical point, in which the interplay of such factors as the level of noise, the tendency to follow the direction and the acceleration of others results in large fluctuations. Analysis: We have proposed a novel link-density based approach to finding overlapping communities in large networks. The algorithm used for the implementation of this technique is very efficient for most real networks, and provides full statistics quickly. Correspondingly, we have developed a by now popular, user-friendly, freely downloadable software for finding overlapping communities. Extending our method to the time-dependent regime, we found that large groups in evolving networks persist for longer if they are capable of dynamically altering their membership, thus, an ability to change the group composition results in better adaptability. We also showed that knowledge of the time commitment of members to a given community can be used for estimating the community's lifetime
Weighted network modules
The inclusion of link weights into the analysis of network properties allows
a deeper insight into the (often overlapping) modular structure of real-world
webs. We introduce a clustering algorithm (CPMw, Clique Percolation Method with
weights) for weighted networks based on the concept of percolating k-cliques
with high enough intensity. The algorithm allows overlaps between the modules.
First, we give detailed analytical and numerical results about the critical
point of weighted k-clique percolation on (weighted) Erdos-Renyi graphs. Then,
for a scientist collaboration web and a stock correlation graph we compute
three-link weight correlations and with the CPMw the weighted modules. After
reshuffling link weights in both networks and computing the same quantities for
the randomised control graphs as well, we show that groups of 3 or more strong
links prefer to cluster together in both original graphs.Comment: 19 pages, 7 figure
Detecting communities using asymptotical Surprise
Nodes in real-world networks are repeatedly observed to form dense clusters,
often referred to as communities. Methods to detect these groups of nodes
usually maximize an objective function, which implicitly contains the
definition of a community. We here analyze a recently proposed measure called
surprise, which assesses the quality of the partition of a network into
communities. In its current form, the formulation of surprise is rather
difficult to analyze. We here therefore develop an accurate asymptotic
approximation. This allows for the development of an efficient algorithm for
optimizing surprise. Incidentally, this leads to a straightforward extension of
surprise to weighted graphs. Additionally, the approximation makes it possible
to analyze surprise more closely and compare it to other methods, especially
modularity. We show that surprise is (nearly) unaffected by the well known
resolution limit, a particular problem for modularity. However, surprise may
tend to overestimate the number of communities, whereas they may be
underestimated by modularity. In short, surprise works well in the limit of
many small communities, whereas modularity works better in the limit of few
large communities. In this sense, surprise is more discriminative than
modularity, and may find communities where modularity fails to discern any
structure
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