Some basic aspects of the recently discovered phenomenon of local immunodeficiency [1] generated by antigenic cooperation in cross-immunoreactivity (CR) networks are investigated. We prove that local immunodeficiency (LI) that's stable under perturbations already occurs in very small networks and under general conditions on their parameters. Therefore our results are applicable not only to Hepatitis C where CR networks are known to be large [1], but also to other diseases with CR. A major necessary feature of such networks is the non-homogeneity of their topology. It is also shown that one can construct larger CR networks with stable LI by using small networks with stable LI as their building blocks. Our results imply that stable LI occurs in networks with quite general topology. In particular, the scale-free property of a CR network, assumed in [1], is not required.

No abstract

In the recently developed theory of isospectral transformations of networks isospectral compressions are performed with respect to some chosen characteristic (attribute) of nodes (or edges) of networks. Each isospectral compression (when a certain characteristic is fixed) defines a dynamical system on the space of all networks. It is shown that any orbit of such dynamical system which starts at any finite network (as the initial point of this orbit) converges to an attractor. Such attractor is a smaller network where a chosen characteristic has the same value for all nodes (or edges). We demonstrate that isospectral contractions of one and the same network defined by different characteristics of nodes (or edges) may converge to the same as well as to different attractors. It is also shown that spectrally equivalent with respect to some characteristic networks could be non-spectrally equivalent for another characteristic of nodes (edges). These results suggest a new constructive approach to analysis of networks structures and to comparison of topologies of different networks. Mathematics Subject Classification: 05C50, 15A18

Feedbacks between strategies and the environment are common in social-ecological, evolutionary ecological and even psychological-economic systems. Using common resources is always a dilemma for community members, like the tragedy of the commons. Here, we consider replicator dynamics with feedback-evolving games, where the pay-offs switch between two different matrices. Although each pay-off matrix on its own represents an environment where cooperators and defectors cannot coexist stably, we show that it is possible to design appropriate switching control laws and achieve persistent oscillations of strategy abundance. This result should help guide the widespread problem of population state control in microbial experiments and other social problems with eco-evolutionary feedback loops.

Isospectral transformations (IT) of matrices and networks allow for compression of either object while keeping all the information about their eigenvalues and eigenvectors.We analyze here what happens to generalized eigenvectors under isospectral transformations and to what extent the initial network can be reconstructed from its compressed image under IT. We also generalize and essentially simplify the proof that eigenvectors are invariant under isospectral transformations and generalize and clarify the notion of spectral equivalence of networks. Mathematics Subject Classification: 05C50, 15A18of spectrally equivalent matrices and networks. Particularly it is demonstrated that there are essential differences between the standard notion of isospectral matrices and spectral equivalence of networks. A new proof of the preservation of eigenvectors under ITs is given which is shorter and applicable to a more general situation than the one in [2]. Isospectral Graph ReductionsIn this section we recall definitions of the isospectral transformations of graphs and networks.Let W be the set of rational functions of the form w(λ)are polynomials having no common linear factors, i.e., no common roots, and where q(λ) is not identically zero. W is a field under addition and multiplication [1].Let G be the class of all weighted directed graphs with edge weights in W. More precisely, a graph G ∈ G is an ordered tripleDenote by M G = (w(i, j)) i, j∈V the weighted adjacency matrix of G, with the convention that w(i, j) = 0 whenever (i, j) E. We will alternatively refer to graphs as networks because weighted adjacency matrices define all static (i.e. non evolving) real world networks.Observe that the entries of M G are rational functions. Let's write M G (λ) instead of M G here to emphasize the role of λ as a variable. For M G (λ) ∈ W n×n , we define the spectrum, or multiset of eigenvalues to beNotice that σ(M G (λ)) can have more than n elements, some of which can be the same.Throughout the rest of the paper, the spectrum is understood to be a set that includes multiplicities. The element α of the multiset A has multiplicity m if there are m elements of A equal to α. If α ∈ A with multiplicity m and α ∈ B with multiplicity n, then (i) the union A ∪ B is a multiset in which α has multiplicity m + n; and (ii) the difference A − B is a multiset in which α has multiplicity m − n if m − n > 0 and where α A − B otherwise.Similarly, the multiset A ⊂ B means for any α ∈ A, we have α ∈ B, and the multiplicity of α in A, is less than or equal to the mutliplicity of α in B.An eigenvector for eigenvalue λ 0 ∈ σ(M G (λ)) is defined to be u ∈ C n , u 0 such thatOne can see the eigenvectors of M G (λ) ∈ W n×n for λ 0 are the same as the eigenvectors of M G (λ 0 ) ∈ C n×n for λ 0 . Similarly the generalized eigenvectors of M G (λ) for λ 0 are the generalized eigenvectors of M G (λ 0 ) for λ 0 . A path γ = (i 0 , . . . , i p ) in the graph G = (V, E, w) is an ordered sequence of distinct vertices i 0 , . . . , i p ∈ V such that (i l , i l+1 ) ∈ E ...